Volume form

 ( Determinants ) 

In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold $M$ of dimension $n$, a volume form is an $n$-form. It is an element of the space of sections of the line bundle $\backslash textstyle\{\backslash bigwedge\}^n(T^*M)$, denoted as $\backslash Omega^n(M)$. A manifold admits a nowhere-vanishing volume form if and only if it is orientable. An orientable manifold has infinitely many volume forms, since multiplying a volume form by a nowhere-vanishing real valued function yields another volume form. On non-orientable manifolds, one may instead define the weaker notion of a density.

A volume form provides a means to define the integral of a function on a differentiable manifold. In other words, a volume form gives rise to a measure with respect to which functions can be integrated by the appropriate Lebesgue integral. The absolute value of a volume form is a volume element, which is also known variously as a twisted volume form or pseudo-volume form. It also defines a measure, but exists on any differentiable manifold, orientable or not.

Kähler manifolds, being , are naturally oriented, and so possess a volume form. More generally, the $n$th exterior power of the symplectic form on a symplectic manifold is a volume form. Many classes of manifolds have canonical volume forms: they have extra structure which allows the choice of a preferred volume form. Oriented pseudo-Riemannian manifolds have an associated canonical volume form.

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Orientation The following will only be about orientability of differentiable manifolds (it's a more general notion defined on any topological manifold).

A manifold is orientable if it has a coordinate atlas all of whose transition functions have positive Jacobian determinants. A selection of a maximal such atlas is an orientation on $M.$ A volume form $\backslash omega$ on $M$ gives rise to an orientation in a natural way as the atlas of coordinate charts on $M$ that send $\backslash omega$ to a positive multiple of the Euclidean volume form $dx^1\; \backslash wedge\; \backslash cdots\; \backslash wedge\; dx^n.$

A volume form also allows for the specification of a preferred class of moving frame on $M.$ Call a basis of tangent vectors $(X\_1,\; \backslash ldots,\; X\_n)$ right-handed if $$\backslash omega\backslash left(X\_1,\; X\_2,\; \backslash ldots,\; X\_n\backslash right)\; >\; 0.$$

The collection of all right-handed frames is acted upon by the group $\backslash mathrm\{GL\}^+(n)$ of general linear mappings in $n$ dimensions with positive determinant. They form a Principal bundle of the linear frame bundle of $M,$ and so the orientation associated to a volume form gives a canonical reduction of the frame bundle of $M$ to a sub-bundle with structure group $\backslash mathrm\{GL\}^+(n).$ That is to say that a volume form gives rise to G-structure on $M.$ More reduction is clearly possible by considering frames that have

Thus a volume form gives rise to an $\backslash mathrm\{SL\}(n)$-structure as well. Conversely, given an $\backslash mathrm\{SL\}(n)$-structure, one can recover a volume form by imposing () for the special linear frames and then solving for the required $n$-form $\backslash omega$ by requiring homogeneity in its arguments.

A manifold is orientable if and only if it has a nowhere-vanishing volume form. Indeed, $\backslash mathrm\{SL\}(n)\; \backslash to\; \backslash mathrm\{GL\}^+(n)$ is a deformation retract since $\backslash mathrm\{GL\}^+\; =\; \backslash mathrm\{SL\}\; \backslash times\; \backslash R^+,$ where the positive reals are embedded as scalar matrices. Thus every $\backslash mathrm\{GL\}^+(n)$-structure is reducible to an $\backslash mathrm\{SL\}(n)$-structure, and $\backslash mathrm\{GL\}^+(n)$-structures coincide with orientations on $M.$ More concretely, triviality of the determinant bundle $\backslash Omega^n(M)$ is equivalent to orientability, and a line bundle is trivial if and only if it has a nowhere-vanishing section. Thus, the existence of a volume form is equivalent to orientability.

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Relation to measures Given a volume form $\backslash omega$ on an oriented manifold, the density $|\backslash omega|$ is a volume Pseudotensor on the nonoriented manifold obtained by forgetting the orientation. Densities may also be defined more generally on non-orientable manifolds.

Any volume pseudo-form $\backslash omega$ (and therefore also any volume form) defines a measure on the by $$\backslash mu\_\backslash omega(U)\; =\; \backslash int\_U\backslash omega\; .$$

The difference is that while a measure can be integrated over a (Borel) subset, a volume form can only be integrated over an oriented cell. In single variable calculus, writing $\backslash int\_b^a\; f\backslash ,dx\; =\; -\backslash int\_a^b\; f\backslash ,dx$ considers $dx$ as a volume form, not simply a measure, and $\backslash int\_b^a$ indicates "integrate over the cell $a,b$ with the opposite orientation, sometimes denoted $\backslash overline\{a,\}$".

Further, general measures need not be continuous or smooth: they need not be defined by a volume form, or more formally, their Radon–Nikodym derivative with respect to a given volume form need not be absolutely continuous.

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Divergence Given a volume form $\backslash omega$ on $M,$ one can define the divergence of a vector field $X$ as the unique scalar-valued function, denoted by $\backslash operatorname\{div\}\; X,$ satisfying $$(\backslash operatorname\{div\}\; X)\backslash omega\; =\; L\_X\backslash omega\; =\; d(X\; \backslash mathbin\{\backslash !\backslash rfloor\}\; \backslash omega)\; ,$$ where $L\_X$ denotes the Lie derivative along $X$ and $X\; \backslash mathbin\{\backslash !\backslash rfloor\}\; \backslash omega$ denotes the interior product or the left contraction of $\backslash omega$ along $X.$ If $X$ is a Compact support vector field and $M$ is a manifold with boundary, then Stokes' theorem implies $$\backslash int\_M\; (\backslash operatorname\{div\}\; X)\backslash omega\; =\; \backslash int\_\{\backslash partial\; M\}\; X\; \backslash mathbin\{\backslash !\backslash rfloor\}\; \backslash omega,$$ which is a generalization of the divergence theorem.

The solenoidal vector fields are those with $\backslash operatorname\{div\}\; X\; =\; 0.$ It follows from the definition of the Lie derivative that the volume form is preserved under the Vector flow of a solenoidal vector field. Thus solenoidal vector fields are precisely those that have volume-preserving flows. This fact is well-known, for instance, in fluid mechanics where the divergence of a velocity field measures the compressibility of a fluid, which in turn represents the extent to which volume is preserved along flows of the fluid.

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Special cases

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Lie groups For any Lie group, a natural volume form may be defined by translation. That is, if $\backslash omega\_e$ is an element of $\{\backslash textstyle\backslash bigwedge\}^n\; T\_e^*G,$ then a left-invariant form may be defined by $\backslash omega\_g\; =\; L\_\{g^\{-1\}\}^*\backslash omega\_e,$ where $L\_g$ is left-translation. As a corollary, every Lie group is orientable. This volume form is unique up to a scalar, and the corresponding measure is known as the Haar measure.

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Symplectic manifolds Any symplectic manifold (or indeed any almost symplectic manifold) has a natural volume form. If $M$ is a $2\; n$-dimensional manifold with symplectic form $\backslash omega,$ then $\backslash omega^n$ is nowhere zero as a consequence of the nondegeneracy of the symplectic form. As a corollary, any symplectic manifold is orientable (indeed, oriented). If the manifold is both symplectic and Riemannian, then the two volume forms agree if the manifold is Kähler.

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Riemannian volume form Any oriented pseudo-Riemannian (including Riemannian) manifold has a natural volume form. In local coordinates, it can be expressed as $$\backslash omega\; =\; \backslash sqrt$$ dx^1\wedge \dots \wedge dx^n where the $dx^i$ are 1-forms that form a positively oriented basis for the cotangent bundle of the manifold. Here, $|g|$ is the absolute value of the determinant of the matrix representation of the metric tensor on the manifold.

The volume form is denoted variously by $$\backslash omega\; =\; \backslash mathrm\{vol\}\_n\; =\; \backslash varepsilon\; =\; \{\backslash star\}(1).$$

Here, the $\{\backslash star\}$ is the Hodge star, thus the last form, $\{\backslash star\}\; (1),$ emphasizes that the volume form is the Hodge dual of the constant map on the manifold, which equals the Levi-Civita tensor $\backslash varepsilon.$

Although the Greek letter $\backslash omega$ is frequently used to denote the volume form, this notation is not universal; the symbol $\backslash omega$ often carries many other meanings in differential geometry (such as a symplectic form).

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Invariants of a volume form Volume forms are not unique; they form a torsor over non-vanishing functions on the manifold, as follows. Given a non-vanishing function $f$ on $M,$ and a volume form $\backslash omega,$ $f\backslash omega$ is a volume form on $M.$ Conversely, given two volume forms $\backslash omega,\; \backslash omega\text{'},$ their ratio is a non-vanishing function (positive if they define the same orientation, negative if they define opposite orientations).

In coordinates, they are both simply a non-zero function times Lebesgue measure, and their ratio is the ratio of the functions, which is independent of choice of coordinates. Intrinsically, it is the Radon–Nikodym derivative of $\backslash omega\text{'}$ with respect to $\backslash omega.$ On an oriented manifold, the proportionality of any two volume forms can be thought of as a geometric form of the Radon–Nikodym theorem.

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No local structure A volume form on a manifold has no local structure in the sense that it is not possible on small open sets to distinguish between the given volume form and the volume form on Euclidean space . That is, for every point $p$ in $M,$ there is an open neighborhood $U$ of $p$ and a diffeomorphism $\backslash varphi$ of $U$ onto an open set in $\backslash R^n$ such that the volume form on $U$ is the pullback of $dx^1\backslash wedge\backslash cdots\backslash wedge\; dx^n$ along $\backslash varphi.$

As a corollary, if $M$ and $N$ are two manifolds, each with volume forms $\backslash omega\_M,\; \backslash omega\_N,$ then for any points $m\; \backslash in\; M,\; n\; \backslash in\; N,$ there are open neighborhoods $U$ of $m$ and $V$ of $n$ and a map $f\; :\; U\; \backslash to\; V$ such that the volume form on $N$ restricted to the neighborhood $V$ pulls back to volume form on $M$ restricted to the neighborhood $U$: $f^*\backslash omega\_N\backslash vert\_V\; =\; \backslash omega\_M\backslash vert\_U.$

In one dimension, one can prove it thus: given a volume form $\backslash omega$ on $\backslash R,$ define $$f(x)\; :=\; \backslash int\_0^x\; \backslash omega.$$ Then the standard Lebesgue measure $dx$ pulls back to $\backslash omega$ under $f$: $\backslash omega\; =\; f^*dx.$ Concretely, $\backslash omega\; =\; f\text{'}\backslash ,dx.$ In higher dimensions, given any point $m\; \backslash in\; M,$ it has a neighborhood locally homeomorphic to $\backslash R\backslash times\backslash R^\{n-1\},$ and one can apply the same procedure.

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Global structure: volume A volume form on a connected manifold $M$ has a single global invariant, namely the (overall) volume, denoted $\backslash mu(M),$ which is invariant under volume-form preserving maps; this may be infinite, such as for Lebesgue measure on $\backslash R^n.$ On a disconnected manifold, the volume of each connected component is the invariant.

In symbols, if $f\; :\; M\; \backslash to\; N$ is a diffeomorphism of manifolds that pulls back $\backslash omega\_N$ to $\backslash omega\_M,$ then $$\backslash mu(N)\; =\; \backslash int\_N\; \backslash omega\_N\; =\; \backslash int\_\{f(M)\}\; \backslash omega\_N\; =\; \backslash int\_M\; f^*\backslash omega\_N\; =\; \backslash int\_M\; \backslash omega\_M\; =\; \backslash mu(M)\backslash ,$$ and the manifolds have the same volume.

Volume forms can also be pulled back under , in which case they multiply volume by the cardinality of the fiber (formally, by integration along the fiber). In the case of an infinite sheeted cover (such as $\backslash R\; \backslash to\; S^1$), a volume form on a finite volume manifold pulls back to a volume form on an infinite volume manifold.

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See also

• Poincaré metric provides a review of the volume form on the complex plane

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